Deforming a maze mesh into a hole
Tissue+Solidify

I would personally model a piece of the maze and spread it on a surface with Tissue add-on. Just pick the component first and then the generator and click "tessellate". From the add-on wiki:


  Tessellate tool allows you to copy a selected object (Component) on the faces of the active object (Generator), adapting its bounding box to the shape of quad-faces.




Try to keep the faces of the generator (was made by slicing a torus) as square as possible to avoid distortion. You can improve the component (was made with Hilbert node from Sverchok) to make it more tileable than mine.

Then I'll add a solidify modifier to the geometry only after the spreading to be sure that the walls are perpendicular to the surface.



Finish with positioning the camera slightly upon the walls.